A Displacement Metric for Finite Sets of Rigid Body Displacements

F
T1
T2
PF
Figure 6. Principal Frame for Four Desired Locations
T1
F
T2
T3
T4
T5
Figure 7. Principal Frame for Ten Desired Locations
The distance from the first location to the principal frame was
found to be 2.7488. The distance between location #1 and
location #2 was found to be 0.3485.
CONCLUSIONS
We have developed a metric for a finite set of rigid body
displacements which uses a mapping of the special Euclidean
group SE(N-1). This technique is based on embedding SE(N-
1) into SO(N) via the polar decomposition of the homogeneous
transform representation of SE(N-1). To yield a useful metric for
a finite set of displacements appropriate for design applications,
the principal frame and the characteristic length are used. A bi-
T3
T6
PF
T7
T8
T9
T4
T10
Table 3. Ten Desired Locations.
# x y z Long (θ) Lat (φ) Roll (ψ)
1 1.00 0.00 5.00 100 0.00 0.00
2 2.00 0.00 4.00 90 0.00 10.00
3 3.00 0.00 3.00 80 0.00 20.00
4 4.00 0.00 2.00 70 0.00 30.00
5 5.00 0.00 1.00 60 0.00 40.00
6 6.00 0.00 −1.00 50 0.00 50.00
7 7.00 0.00 −2.00 40 0.00 60.00
8 8.00 0.00 −3.00 30 0.00 70.00
9 9.00 0.00 −4.00 20 0.00 80.00
10 10.00 0.00 −5.00 10 0.00 90.00
invariant metric on SO(N) is then used to measure the distance
between any two displacements in SE(N-1). A detailed algorithm
for the application of this method was presented and illustrated
by three examples. This technique has potential applications in
mechanism synthesis and robot motion planning.
ACKNOWLEDGMENT
This material is based upon work supported by the National
Science Foundation under grant #0422705. Any opinions,
findings, and conclusions or recommendations expressed in this
material are those of the authors and do not necessarily reflect
the views of the National Science Foundation.
REFERENCES
[1] Lin, Q., and Burdick, J., 2000. “Objective and frameinvariant
kinematic metric functions for rigid bodies”. International
Journal for Robotics Research, 19(6), pp. 612–
625.
[2] Park, F., 1995. “Distance metrics on the rigid-body motions
with applications to mechanism design”. ASME Journal of
Mechanical Design, 117(1), September, pp. 48–54.
[3] Kazerounian, K., and Rastegar, J., 1992. “Object norms:
A class of coordinate and metric independent norms for
displacements”. In Proceedings of the ASME 1998 Design
Engineering Technical Conferences and Computers and
Information Conference.
[4] Martinez, J. M. R., and Duffy, J., 1995. “On the metrics
of rigid body displacements for infinite and finite bodies”.
ASME Journal of Mechanical Design, 117(1), pp. 41–47.
[5] Larochelle, P., and McCarthy, J., 1995. “Planar motion
6 Copyright c○ 2008 by ASME